Mathematics for the interested outsider

Integrals over Manifolds (part 1)

We’ve defined how to integrate forms over chains made up of singular cubes, but we still haven’t really defined integration on manifolds. We’ve sort of waved our hands at the idea that integrating over a cube is the same as integrating over its image, but this needs firming up. In particular, we will restrict to oriented manifolds.

To this end, we start by supposing that an -form is supported in the image of an orientation-preserving singular -cube . Then we will define

Indeed, here the image of is some embedded submanifold of that even agrees with its orientation. And since is zero outside of this submanifold it makes sense to say that the integral over the submanifold — over the singular cube — is the same as the integral over the whole manifold.

What if we have two orientation-preserving singular cubes and that both contain the support of ? It only makes sense that they should give the same integral. And, indeed, we find that

where we use to reparameterize our integral. Of course, this function may not be defined on all of , but it’s defined on , where is supported, and that’s enough.

[…] Remembering that diffeomorphism is meant to be our idea of what it means for two smooth manifolds to be “equivalent”, this means that is either equivalent to or to . And I say that this equivalence comes out in integrals. […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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